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Computer Science, Symbolic Computation, cs.SC
Abstract:
A zero-dimensional polynomial ideal may have a lot of complex zeros. But
sometimes, only some of them are needed. In this paper, for a zero-dimensional
ideal $I$, we study its complex zeros that locate in another variety
$\textbf{V}(J)$ where $J$ is an arbitrary ideal.
The main problem is that for a point in $\textbf{V}(I) \cap
\textbf{V}(J)=\textbf{V}(I+J)$, its multiplicities w.r.t. $I$ and $I+J$ may be
different. Therefore, we cannot get the multiplicity of this point w.r.t. $I$
by studying $I + J$. A straightforward way is that first compute the points of
$\textbf{V}(I + J)$, then study their multiplicities w.r.t. $I$. But the former
step is difficult to realize exactly.
In this paper, we propose a natural geometric explanation of the localization
of a polynomial ring corresponding to a semigroup order. Then, based on this
view, using the standard basis method and the border basis method, we introduce
a way to compute the complex zeros of $I$ in $\textbf{V}(J)$ with their
multiplicities w.r.t. $I$. As an application, we compute the sum of Milnor
numbers of the singular points on a polynomial hypersurface and work out all
the singular points on the hypersurface with their Milnor numbers.
